Fully plastic failure stresses for circumferentially surface cracked pipes subjected to tensile loading can be estimated by means of limit load criteria based on the net-section stress approach. Limit load criteria of the first type (labelled LLC-1) were derived from the balance of uniaxial forces. Limit load criteria of the second type are given in Section XI of the ASME (American Society of Mechanical Engineering) Code, and were derived from the balance of bending moment and axial force. These are labelled LLC-2. Fully plastic failure stresses estimated by using LLC-1 and LLC-2 were compared. The stresses estimated by LLC-1 are always larger than those estimated by LLC-2. From the literature survey of experimental data, failure stresses obtained by both types of LLC were compared with the experimental data. It can be stated that failure stresses calculated by LLC-1 are better than those calculated by LLC-2 for shallow cracks. On the contrary, for deep cracks, LLC-2 predictions of failure stresses are fairly close to the experimental data. Furthermore, allowable circumferential crack sizes obtained by LLC-1 were compared with the sizes given in Section XI of the ASME Code. The allowable crack sizes obtained by LLC-1 are larger than those obtained by LLC-2. It can be stated that the allowable crack size for tensile stress depends on the condition of constraint of the pipe, and the allowable cracks given in Section XI of the ASME Code are conservative.